Abstract
The consideration of material losses in phononic crystals leads naturally to the introductionof complex valued eigenwavevectors or eigenfrequencies representing the attenuation of elastic wavesin space or in time, respectively. Here, we propose a new technique to obtain phononic band structureswith complex eigenfrequencies but real wavevectors, in the case of viscoelastic materials, wheneverelastic losses are proportional to frequency. Complex-eigenfrequency band structures are obtainedfor a sonic crystal in air, and steel/epoxy and silicon/void phononic crystals, with realistic viscouslosses taken into account. It is further found that the imaginary part of eigenfrequencies are wellpredicted by perturbation theory and are mostly independent of periodicity, i.e., they do not accountfor propagation losses but for temporal damping of Bloch waves.
Highlights
Phononic crystals for elastic waves are classical analogs to crystal lattices for phonons [1,2].They are described with continuum mechanics and can be viewed as periodic composites
Bloch waves of phononic crystals are obtained by solving an eigenfrequency problem under periodic boundary conditions
For the 2D hexagonal lattice phononic crystal of cylindrical steel rods in an epoxy matrix depicted in Figure 2a, the complex-eigenfrequency band structure shown in Figure 2b as a large
Summary
Phononic crystals for elastic waves are classical analogs to crystal lattices for phonons [1,2]. E + ıωη with ω being the angular frequency and η the viscosity, applies well at ultrasonic frequencies In crystals such as silicon or quartz, material loss can often be modeled by adding an imaginary part to the elastic tensor, in a first approximation proportional to the frequency [3,4]. Sonic crystals for acoustic waves contain viscous fluids that can be described by a dynamical viscosity leading to a bulk modulus whose imaginary part increases proportionally to frequency. This applies to common fluids supporting sound propagation, such as air and water. The complex-eigenfrequency band structure ω (k ) does not capture any spatial propagation information but provides one with information complementary to the complex band structure k (ω )
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